vortex ﬁlamentation: its onset and its role on...

Dynamics of Atmospheres and Oceans40 (2005) 23–42

Vortex filamentation: its onset and its role onaxisymmetrization and merger

O.U. Velasco Fuentes∗

Departamento de Oceanograf´ıa Fısica, CICESE, Ensenada, Baja California, M´exico

Received 23 February 2004; accepted 4 October 2004Available online 2 December 2004

Abstract

Vortex filamentation in two-dimensional flows is revisited. Attention is centered on its onset andits role on the axisymmetrization of elongated vortices and on the merger of vortex pairs. The objec-tive is to test two generally accepted hypotheses; namely, that filamentation occurs because a saddlestagnation point enters the vortex and that filaments cause both axisymmetrization and merger. Thesehypotheses are based on the analysis of the Eulerian flow geometry, i.e., the set of stagnation pointsand associated streamlines of the instantaneous velocity field. Here, on the contrary, filamentationis described and quantified by analyzing the Lagrangian flow geometry, i.e., the set of hyperbolictrajectories and associated manifolds of the time evolving velocity field. This dynamical-systemsapproach is applied to the numerical representation of the velocity field obtained by solving the Eulerequations with a vortex-in-cell model. Filamentation is found to occur because a stable manifoldof a hyperbolic trajectory enters the vortex, and it is unimportant whether the hyperbolic trajectoryor the saddle point are inside the vortex or whether they are outside. It is also found that filamen-tation, although an important part of both axisymmetrization and merger, is not the cause of theseprocesses.© 2004 Elsevier B.V. All rights reserved.

Keywords:Vortex filamentation; Axisymmetrization; Merger; Lobe dynamics

∗ Present address: CICESE, P.O. Box 434844, San Diego, CA 92143, USA. Tel.: + 52 646 1750500;fax: + 52 646 1750547.

E-mail address:[email protected] (O.U. Velasco Fuentes).

0377-0265/$ – see front matter© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.dynatmoce.2004.10.003

24 O.U. Velasco Fuentes / Dynamics of Atmospheres and Oceans 40 (2005) 23–42

1. Introduction

Vortex axisymmetrization (the relaxation of elongated vortices towards axial symmetry)and vortex merger (the coalescence of a pair of like-signed vortices into a single structure) arefundamental phenomena in two-dimensional flows and share important features(Melanderet al., 1987, 1988). It is well known that in the course of these processes the vorticesexpel filaments in amounts that depend on the initial conditions (vorticity distribution,aspect ratio of the elongated vortex, intercentroid distance of the vortex pair, etc.). Vortexfilamentation is generally considered to occur because a stagnation point of saddle type lieswithin the vortex(Melander et al., 1987, 1988; Polvani et al., 1989; Mariotti et al., 1994;Cerretelli and Williamson, 2003)and it has been proposed as the cause of axisymmetrization(Melander et al., 1987)and merger(Melander et al., 1988; Cerretelli and Williamson, 2003).These hypotheses are based on comparing the vorticity distribution with the Eulerian flowgeometry, which may be defined as the set of stagnation points of saddle type and theassociated streamlines as observed in the instantaneous velocity field. One weakness of thisapproach is that it usually requires the computation of the Eulerian geometry in a movingreference system where the flow appears to be stationary. Such a system, however, does notexist for the flows under study here because they are essentially time dependent. At best,systems can be found where the flow appears to evolve slowly, but the particular systemused is always the observer’s choice. SeeFig. 1, which shows the Eulerian flow geometryfor an elliptic vortex and for a pair of vortices in a fixed frame [(a) and (c), respectively] andin a frame rotating with the vortices [(b) and (d), respectively]. Fortunately, the location ofa saddle point does not change significantly with any reasonably chosen reference systemand this small uncertainty may be considered a minor weakness of the approach. A crucialflaw is, however, that filamentation usually starts well before any saddle point enters thevortex.

Vortex filamentation is a manifestation of fluid advection. Therefore, it will be studiedhere using methods from the theory of transport in dynamical systems. In particular, we willuse the Lagrangian flow geometry, i.e., the set of hyperbolic trajectories and their associatedmanifolds. Loosely speaking, the hyperbolic trajectory is the path of a fluid particle whichattracts one set of particles and repels a different set of particles. Since the hyperbolictrajectory is the path of a material particle, we will refer to this particle as hyperbolicparticle. The set of particles, which are attracted towards the hyperbolic particle constitutethe stable manifold, and the set of repelled particles constitute the unstable manifold. Here,of course, the terms attract and repel are used only as a description of how the fluid movesin the vicinity of the hyperbolic particle.

The relation between the Lagrangian and Eulerian geometries depends on the way theflow under consideration evolves in time. In steady flows both geometries are identical: thesaddle points are hyperbolic trajectories, and the stretching and squeezing streamlines arethe unstable and stable manifolds, respectively. In flows that can be considered as a meansteady flow plus a small time-dependent perturbation, the two geometries closely resembleeach other: the hyperbolic particle is always close to the saddle point and the manifoldsemanating from the hyperbolic trajectories are parallel to the streamlines in the vicinity ofthe saddle point. Early studies of chaotic advection dealt mostly with flows of this type (e.g.,Rom-Kedar et al., 1990). Realistic flows are usually more complicated: they are aperiodic

O.U. Velasco Fuentes / Dynamics of Atmospheres and Oceans 40 (2005) 23–42 25

Fig. 1. Stream function of a single elliptic vortex (a and b) and a pair of equal circular vortices (c and d), asobserved in a fixed reference frame (a and c) and in a frame co-rotating with the vortex structure (b and d). Thegray areas are the vortices, the black dots indicate the saddle points, and the arrows indicate the direction of particlemotion along the streamlines.

and may undergo qualitative changes in the course of the evolution. For instance, vortexmerger implies the disappearance of one of the three saddle stagnation points present inthe initial condition, and complete axisymmetrization implies the disappearance of the twosaddle points. As a consequence the Eulerian flow geometry may differ significantly fromthe Lagrangian one, and any consequences derived from the former may be misleading.

The existence of stagnation points may facilitate the detection of the Lagrangian flowgeometry. Some analytic results guarantee that, in periods of slow evolution, hyperbolic tra-jectories exist in the neighborhood of the stagnation points(Haller and Poje, 1998; Malhotraand Wiggins, 1998). The Eulerian geometry is thus used as a first step in the constructionof the Lagrangian geometry. This approach has been very successful for computing trans-port templates for vortex–jet and vortex–vortex interaction problems (e.g.,Poje and Haller,1999; Velasco Fuentes, 2001). Therefore, this approach is best suited for achieving the ob-jectives of the present study: (a) to relate the Eulerian and the Lagrangian flow geometriesto the onset of filamentation and (b) to determine the role that filaments play in vortexaxisymmetrization and vortex merger.


The rest of the paper is organized as follows. A brief description of the methods is given inSection2. In Section3we study the axisymmetrization of elliptical vortices with nonuniformvorticity profile. Section4 deals with the merger of vortex pairs, and the conclusions aregiven in Section5.

2. Methods

2.1. Numerical solution of the equation of motion

We will consider the two-dimensional, unforced flow of an ideal fluid. Such system isgoverned by the Euler equation, which can be written in the following form:

∂ω

∂t+ J(ω,ψ) = 0, (1)

whereω = ∂v/∂x− ∂u/∂y is the vorticity, andψ is the stream function. Eq.(1) is solvedusing the vortex-in-cell method (VIC) as implemented inVelasco Fuentes (2001)(see alsoHockney and Eastwood (1981), for a general account of particle methods). The numericaldomain is a square, uniform grid of 256× 256 points with free-slip boundary conditions.In the simulations of vortex axisymmetrization there are 12 grid points per mean vortexradius, and the elliptic patch of vorticity is represented by about 80,000 point vortices.In the simulations of vortex merger there are ten grid points per vortex radius, and eachcircular patch of vorticity is represented by about 35,000 point vortices. Some calculationsregarding the accuracy of the simulations may be found in theAppendix A.

2.2. Computation of particle trajectories

Particle trajectories are the solutions of the following pair of ordinary differential equa-tions:

dx

dt= ∂ψ(x, y, t)

∂y,

dy

dt= −∂ψ(x, y, t)

∂x, (2)

whereψ is the stream function obtained by numerically solving Eq.(1) with the VICmethod. Hence, Eq.(2) must be integrated when the right-hand side is known at discretespatial and temporal grid points only. The database consists ofN + 1 slicesof data, wherethenth slice is the solution computed at timetn = nt, for n = 0,1, . . . , N. Each sliceis a 2D array of data defining the stream function onM ×M grid points. In this study,N = 1000,t ≈ τ/125 (whereτ is the eddy turnover time), andM = 256.

The integration of Eq.(2) thus requires interpolation in three dimensions to find thevelocity at arbitrary points (x, y, t). For consistency, the interpolation in space is madewith the same biquadratic scheme used in the VIC model. The time integration is madewith a second-order Runge–Kutta scheme, therefore, even thoughψ is output at ev-ery time step available in the VIC model, the data slices are interpolated by the time-integrator.


2.3. Finding hyperbolic trajectories

Since the flow is essentially time dependent, the relation between the trajectories offluid particles (Lagrangian dynamics) and the geometry of the instantaneous velocity field(Eulerian flow) is not straightforward. We expect, however, that if the saddle point existslong enough and the velocity field around it changes slowly then there is a hyperbolicparticle in its neighborhood. This can be proved rigorously; seeHaller and Poje (1998)andMalhotra and Wiggins (1998)for the theory, andVelasco Fuentes (2001)for an exampleof the use of the criteria. Usually this condition is satisfied uninterruptedly during vortexaxisymmetrization, and it is satisfied at various stages during vortex merger. When thecondition is satisfied, the position of the hyperbolic particle and its manifolds at any giventimet can be computed by the method described below. With suitable variations, this methodhas been extensively used for velocity fields defined analytically or given as data sets, andwith periodic, quasi-periodic or arbitrary time dependence (e.g.,Rom-Kedar et al., 1990;Beigie et al., 1994; Malhotra and Wiggins, 1998; Velasco Fuentes, 2001).

The starting point of the analysis is the stream functionψ computed by the VIC model(e.g.,Fig. 1a and c). The stream functionΨ observed in a system that rotates with the vorticitydistribution is given by the simple transformationΨ = ψ + 1

2Ω[(x− xc)2 + (y − yc)2

],

whereΩ is the angular velocity and (xc, yc) is the observed center of rotation. Because ofthe initial symmetry, the center of rotation remains at the vortex center when there is onevortex and at the middle point between the vortices when there are two of them. A goodestimation ofΩ is obtained making any of the following assumptions: (a) a single, elongatedvortex is taken to be an elliptic patch of uniform vorticity; or (b) a pair of vortices is takento be a pair of point vortices. In either case the angular velocityΩ is evaluated from thecorresponding analytic expression: (a)Ω = abω/(a+ b)2, whereω is the relative vorticityanda andb the semi-axes of the ellipse; or (b)Ω = Γ/(πd2), whereΓ is the circulation ofeach vortex andd is the intercentroid distance.

The second step is to determine the geometry of the co-rotating stream functionΨ ; thatis to say, locate the saddle points and the streamlines associated to them. A wide varietyof methods and tools for obtaining these geometric elements from a numerically generatedflow field are available (seeVelasco Fuentes, 2001for the implementation used here).

Finally, the stable manifold is obtained computing the evolution, from timet +t totime t, of a short line, which crosses the saddle point ofΨ (x, y, t +t) in the squeezingdirection; and the unstable manifold is obtained computing the evolution, from timet −tto time t, of a short line which crosses the saddle point ofΨ (x, y, t −t) in the stretchingdirection (seeFig. 2). The position of the hyperbolic particle is given by the intersection ofthe manifolds.

3. Evolution of elongated vortices

In the last three decades much work has been done on the evolution of single, unsteadyvortices. Various settings have been considered: a perturbed Kirchhoff’s vortex(Polvani etal., 1989), nonuniform elliptical vortices(Melander et al., 1987; Koumoutsakos, 1997), andvortices with various vorticity distributions embedded in a background flow(Mariotti et


Fig. 2. Numerical computation of the manifolds at timet. Small line segments are placed along the stretching andsqueezing directions of the saddle point at timest −t andt +t, respectively. The evolution of this segmentstoward timet give the unstable and stable manifolds. The intersection of the segments give the position of thehyperbolic particle.

al., 1994). Filamentation of the vortex has usually been observed, sometimes accompaniedby a tendency towards axial symmetry. It has been proposed that filamentation occursbecause there is a saddle point on the boundary or inside the vortex, and that the filamentsdrive the evolution towards axial symmetry(Melander et al., 1987). In this section, wewill test this hypothesis for Kirchhoff vortices and for elliptic vortices with nonuniformvorticity distributions.

3.1. Filamentation of a perturbed Kirchhoff ’s vortex

Fig. 3 shows the evolution of a Kirchhoff vortex in the unstable regime (initial as-pect ratio is 3.33). A perturbation has been prescribed as follows: the vortex contourhas a deformation of wave number 3 and an amplitude of 2% relative to the length ofthe mean radius.Polvani et al. (1989)used the same conditions in their study of theonset of filamentation. The instability produces the ejection of a thin vortex filamentand a slight change of shape but there is no tendency towards axial symmetry. In fact,the vortex is less axisymmetric in the last stage shown than in its initial condition (seecolumn a).

During the whole evolution the Eulerian and the Lagrangian flow geometries (columns band c, respectively) show a remarkable resemblance. Consequently, the hyperbolic particleand the saddle point are closely located, as it is shown inFig. 4. Such a high similarity


Fig. 3. Time evolution of a perturbed Kirchhoff vortex (aspect ratio: 3.3; perturbation wave number: 3; pertur-bation amplitude: 2%). (a) Vortex contour, (b) co-rotating stream function, and (c) stable and unstable mani-folds (thin and thick lines, respectively). The timet∗ is given in units of the rotation period of the unperturbedvortex.


Fig. 4. Path of the saddle point (open circles) and trajectory of the hyperbolic particle (filled circles) during theevolution of the Kirchhoff vortex shown inFig. 3. The size of the circles increases with the time elapsed.

between the two geometries is a consequence of the flow being alwayscloseto a steadysolution of the equations of motion.

Subtle differences exist, however, and these are important in relation to the hypothesisthat we want to test.Fig. 5 shows a detail of the vortex and the corresponding Eulerianand Lagrangian flow geometries when the saddle point enters the vortex. At this momentthe filamentation of the vortex is well under way; i.e., it started when the saddle point wasstill out of the vortex. Notice also that the vortex contour runs fairly parallel to the unstablemanifold, showing that filamentation is governed by the Lagrangian geometry.

3.2. Axisymmetrization of nonuniform vortices

A systematic analysis of the filamentation was performed with vortices of vorticitydistributions taken from the following family

ω =ω0

[1 − (

ra

)n] for r < a,

0 forr > a,(3)

whereω0 is the peak value of the vorticity,r the distance to the vortex centre andn is aninteger. The vorticity profiles considered here are (a)n = 6 (steep) and (b)n = 2 (smooth).The initial aspect ratio (e0) of the vortices was varied in the range 1–3, which correspondsto the stable regime of Kirchhoff vortices. Note, however, that the vortices given by(3)havenonuniform vorticity (except forn = ∞), and they are unsteady for alle0 = 1.

In each numerical experiment we determined the locations of hyperbolic particlesH(which are intrinsic properties of the flow, and therefore frame independent) and of saddlestagnation pointsS(which depend on the choice of co-moving frame). In addition, we com-puted two diagnostic parameters for the approximately elliptic patch of vorticity that survivesthe filamentation process: area (A∗) and aspect ratio (e∗). Fig. 6shows results for vortices


Fig. 5. Detail of the Eulerian and Lagrangian flow geometries at timet∗ = 0.62 for the Kirchhoff vortex shownin Fig. 3. This figure shows the hyperbolic particle (black dot) with its manifolds (stable: thin, continuous line;unstable: thick, continuous line); and the saddle point (open circle) with its associated streamlines (broken lines).The gray area is the vortex.

with steep vorticity profile (n = 6); results for vortices with smooth vorticity profile (n = 2)are qualitatively similar.A∗ ande∗ are the quasi-steady values measured at a late stage (i.e.,when the vortex ceased to undergo noticeable changes) and are given as a fraction of the cor-responding value in the initial condition. Take for instance the vortex withe0 = 3. The valuesof A∗ ande∗ are about 0.8 and 0.65, respectively, which means that at the final stage thevortex preserves only 80% of its original area and its aspect ratio decreases to 0.65× 3 ≈ 2.Fig. 6also shows values ofHandSin the initial condition. The values are scaled by the lengthof the semi-major axis of the vortex, so that values larger than one indicate that the point isoutside the vortex and values lower than one indicate that the point is inside the vortex.

Analyzing the value ofS in combination with the value ofA∗ for a particular initialcondition gives an indication of the influence of the initial location of the saddle pointon the latter occurrence of filamentation.Fig. 6 shows that vortices with 1.5< e0 < 2haveS > 1 (i.e., the saddle point lies outside the vortex) and yetA∗ < 1 (i.e., the areadecreases through filamentation). The same happens with the hyperbolic particle: in theinterval 1.5< e0 < 2.75 the hyperbolic particle lies outside the vortex and yet the areadecreases through filamentation.

In a similar way, analyzing the value ofA∗ in combination with the value ofe∗ fora particular initial condition gives an indication of the relation between filamentation (adecrease inA∗) with axisymmetrization (a decrease ine∗). Fig. 6shows that vortices with1< e0 < 1.5 haveA∗ = 1 (i.e., there is no filamentation) and yete∗ < 1 (i.e., the aspectratio decreased during the evolution).

We may, therefore, conclude the following: (a) the saddle point being inside the vortexis a sufficient condition for filamentation, but it is not a necessary one and (b) filamentationis a sufficient condition for decrease of aspect ratio, but it is not a necessary one.


Fig. 6. Diagnostic parameters for an elliptic vortex of nonuniform vorticity as a function of the initial aspect ratioe0. The figure shows final values of the vortex areaA∗ (filled diamonds) and the aspect ratioe∗ (open squares);and the initial relative position of the hyperbolic particleH (filled circles) and the saddle pointS (open circles).Values ofA∗ ande∗ are relative to their initial values; values ofH andSare relative to the value of the vortexsemi-major axis att∗ = 0 (see text).

4. Merger of pairs of equal vortices

The interaction of two like-signed vortices has been the subject of intense research forthe last three decades. Most studies have dealt with the highly idealized situation of twocircular vortices of equal size and strength. Three flow regimes have been identified in thissetting, depending on the initial vortex separationd∗ = d/R (whered is the intercentroiddistance andR is the vortex radius). These regimes are the following: (a) vortices locatedfar apart circle around each other endlessly without undergoing large deformations, (b)vortices located at an intermediate distance circle around each other while exchangingsome vorticity, and (c) vortices located at a distance smaller than a certain critical valuemerge into a single vortex (e.g.,Melander et al., 1988). The existence of these regimeshas been explained invoking the Eulerian flow geometry(Melander et al., 1988; Cerretelliand Williamson, 2003). According to this hypothesis the evolution depends on how theinitial vorticity field is located with respect to the separatrices of the co-rotating streamfunction (e.g.,Fig. 1d): (a) if the vortices are confined inside the inner, eight-shaped sep-aratrix they rotate around each other without undergoing large deformations or merger;(b) if the vortices overstep the inner separatrix but do not go beyond the middle, peanut-shaped separatrix they rotate around each other, exchange some vorticity, but do not merge;and finally (c) if the vortices extend beyond the middle separatrix they expel filamentsand merge.Melander et al. (1988)proposed this hypothesis simply as an heuristic toolto explain the vortex interaction. Although it has been shown to be inaccurate for variousvorticity distributions(Velasco Fuentes, 2001), the same hypothesis has been recently re-covered and it has been proposed as the physical mechanism for merger(Cerretelli and


Williamson, 2003). Hence we will analyze below the relation of the Eulerian and La-grangian geometries with the different regimes and, especially, the role of filamentation onmerger.

Fig. 7 shows the evolution of a pair of equal vortices in the merger regime (d∗ = 3.0).Column (a) shows the vortex patches, column (b) shows the Eulerian flow geometry asobserved in a co-rotating system and column (c) shows the Lagrangian flow geometry. Acomparison of frames (a) and (b) in the initial condition (t∗ = 0) reveals very clearly thatthe vortices are contained within the middle, peanut-shaped separatrix; therefore, if thehypothesis stated above were correct, they should not merge. The evolution, however, isquite different: the vortices start to merge almost immediately.

Notice that there are large differences between the Eulerian and Lagrangian flow ge-ometries during most of the evolution. These differences arise because at some stage theflow is far from any steady solution of the equations of motion. This far-from-steady stageis the merger event itself, which occurs in a time scale of the order of the eddy turnovertime.

The hyperbolic particle and the saddle point are located close to each other during theearly stages of the evolution (seeFig. 8). But they separate as soon as merger starts (around

Fig. 7. Same asFig. 3but now for a pair of Rankine vortices (initial separationd∗ = 3). The timet∗ is given herein units of the eddy turnover time.


Fig. 7. (Continued).

Fig. 8. Same asFig. 4but now for the pair of Rankine vortices shown inFig 7.


time t∗ = 0.7, given in units of the eddy turnover time) and do not approach each otheragain until the late stages of the evolution (around timet∗ = 3), when the newly formedvortex stabilizes as an elongated vortex.

4.1. Lagrangian geometry and regime transitions

The results of the previous section show that the relative location of the vortices withrespect to the Eulerian flow geometry does not predict the outcome of the interaction. Inthis section, we analyze the Lagrangian flow geometry as the vortices change from oneregime to another in order to see if this geometry provides a clue about the outcome of theinteraction. The following conventions and definitions will facilitate the exposition below.The hyperbolic particles are denoted by a letterp, and a subscript distinguishes each of thethree (0 for the one in the middle, 1 for the one on the left and 2 for the one on the right). Themanifolds are denoted by a letterW, the subscript indicates to which hyperbolic particle itbelongs to, and the superscript indicates if it is stable (s) or unstable (u). The intersectionPof two manifolds is said to be a primary intersection if the manifolds, sayWu

1 fromp1 toPandWs

2 from p2 toP, intersect only atP.Fig. 9 shows the Lagrangian flow geometries for vortices in each regime: (a) elastic-

interaction,d∗ = 3.45; (b) exchange,d∗ = 3.35; and (c) merger,d∗ = 3.25. Only shortsegments of the manifolds are shown so that the flow geometry can be clearly observed.Of particular interest are the primary intersections ofWs

1, with eitherWu1 orWu

2 , that arelocated on the linep1p2 or on the perpendicular to that line atp0. We will refer to theseas symmetric primary intersections (SPI). In the elastic-interaction regime (Fig. 9a) theSPI ofWs

1 with Wu2 is closer top1 than the SPI ofWs

1 with Wu1 (not shown in the figure).

This was to be expected because asd∗ → ∞ the flow becomes steady andWs1 coincides

with Wu2 to form the middle separatrix. The situation changes as the flow enters the ex-

change regime (Fig. 9b), for in this case the SPI ofWs1 with Wu

1 is closer top1 than theSPI ofWs

1 with Wu2 . There is thus a qualitative change in the Lagrangian geometry in

the transition from the elastic-interaction regime to the exchange regime. Finally, in themerger regime,Fig. 9c, we observe the same qualitative behaviour of the manifolds as inthe elastic-interaction regime (N.B. the apparent crossing ofWu

1 andWu2 is the result of

the finite width of the lines: two unstable manifolds can not intersect). Since there is noqualitative difference between the two geometries, it can be concluded that the exchangeregime and the merger regime develop initially in a qualitatively similar way. Note alsothatWs

1 does not enter the vortices; therefore no filamentation is involved in the earlystages of the evolution, which are characterized by a mutual approach of the vortices.This result is confirmed by the time evolution of the minimal distance between the vor-tices and of the length of the vortex contours (Fig. 10). The distance between the vorticesdecreases as soon as the simulation starts, and is reduced to less than 10% of its initialvalue in a timet∗ = 0.5 for vortices with initial separationd∗ = 3.0, and in a timet∗ = 0.8for vortices withd∗ = 3.25. In contrast, the length of the vortex contour grows slowlyat first and rapidly when filamentation starts (aroundt∗ = 0.5 for d∗ = 3.0, and aroundt∗ = 0.8 for d∗ = 3.25). The start of vortex merger thus precedes the start of filamentation(cf. Cerretelli and Williamson (2003), who attributed the vortex approach to filamenta-tion).


Fig. 9. The Lagrangian flow geometry at timet∗ = 0 for a pair of Rankine vortices in (a) the elastic-interactionregime (d∗ = 3.45), (b) the exchange regime (d∗ = 3.35), and (c) the merger regime (d∗ = 3.25). The vortices arerepresented by gray circles; the stable (Ws) and unstable (Wu) manifolds are represented by thin and thick lines,respectively; and the hyperbolic particles (p1 andp2) are indicated by black dots.

4.2. Quantification of vortex filamentation

The Lagrangian flow geometry is essentially time dependent, but the properties of thestable and unstable manifolds enable us to draw conclusions on the basis of the flow geom-etry observed at a given time. In particular, note that the stable and unstable manifolds ofhyperbolic particles are invariant surfaces and particles cannot cross them (e.g.,Malhotraand Wiggins, 1998). Therefore, a patch of fluid located on one side with respect to a man-ifold will remain on that side during the whole evolution. The use of this property in theanalysis of the intersections of the stable manifolds with the vortex enables us to identifyand quantify the areas which are expelled from the vortex.Fig. 11ashows the manifolds ofthe middle hyperbolic particlep0. Particles which are to the right of the manifold (whenlooking in the flow direction indicated by the arrows) are transported towards the othervortex.Fig. 11bshows the manifolds of the hyperbolic particlep1. Particles which are tothe right of the manifold are expelled from the vortex to form the spiral arms.

The amount of area exchanged between the vortices (Ae∗) or detrained in filaments (Ad∗)has been computed for initial conditions in the range 2< d∗ < 3.5 (Fig. 12). This mass,denoted byA∗, is given as a fraction of the initial mass of the vortex. When the vortices are in


Fig. 10. Evolution of the minimal distance between the vortices (continuos line) and the length of the vortexcontour (broken line) in two cases of vortex merger (d∗ = 3.0: thick lines; andd∗ = 3.25: thin lines). Lengths aregiven relative to their initial values, and time is given in units of the eddy turnover time.

touch (d∗ = 2) the detrained areaAd∗ is about 8%; and asd∗ increases,Ad∗ also increases untilit reaches a maximum atd∗ ≈ 2.9, when the vortices loose 20% of their mass. These valuesagree well with the results ofWaugh (1992), who computed the merger efficiency (i.e., thearea that is not detrained) to be about 90% atd∗ = 2 and 78% atd∗ = 2.9. As d∗ furtherincreasesAd∗ decreases until it becomes zero atd∗ ≈ 3.35. Note that for values slightly

Fig. 11. Evaluation of mass transport. The darker gray areas remain with the vortex, the lighter areas are either(a) exchanged with the vortex partner or (b) expelled to the ambient fluid. The stable and unstable manifolds arerepresented by thin and thick lines, respectively, and the hyperbolic points are indicated by black dots.


Fig. 12. Area expelled (continuous line) and exchanged (broken line) by Rankine vortices as a function of theinitial intercentroid distance (d∗). Values are given as a fraction of the total area in the initial vortex.

above the critical distance for merger the amount of mass expelled is small but differentfrom zero. The area exchanged by the vortices (Ae∗) increases from zero atd∗ = 3.4, whichis by definition the boundary between the exchange and the elastic-interaction regimes, tomore than 60% atd∗ = 3.3, the critical distance for merger.

The same calculations were done with the vortices used byMelander et al. (1988):

ω = ω0

1 − exp

(−κ 2R

rexp

(1

r/2R−1

))for r < 2R,

0 forr > 2R,(4)

whereκ = exp(2) log(2)/2 andR is the radius where the vorticity reaches 50% of its peakvalue. This equation gives a vorticity profile with an almost conical shape (except that it issmoothly connected to the extreme values). We have chosen it out of the family given byMelander et al. (1987, 1988)because it was the one they used when applying the Euleriangeometry to explain vortex axisymmetrization and merger. Below we will refer to them asMMZ vortices.

Defining the vortex radius for an arbitrary vorticity profile is not trivial (e.g.,Mitchelland Driscoll, 1996). Natural choices for the profile given by(4)are the radiusrmaxwhere thehighest flow speed is achieved or the radiusr0 where the vorticity becomes zero. Note thatin Eq.(4) rmax/r0 = 0.57, which is substantially smaller than the corresponding values forEq. (3), wherermax/r0 = 1,0.91,0.82, forn = ∞,6,2, respectively. Since we have useda pragmatic definition of vortex filament (a region of a vortex which is expelled from it andsubsequently undergoes stretching in one direction and squeezing in another), the definitionof the vortex radius naturally affects the definition and quantification of the filaments.

If we take rmax as the vortex radius and define “filamentation” as the expulsion offilaments of fluid coming from the regionr < rmax, then there are initial conditions inwhich MMZ vortices merge without expelling any filaments. The maximum amount of mass


Fig. 13. Same asFig. 12, but now for the smooth-profile vortices used byMelander et al. (1988). Thick lines andlarge markers show values obtained whenrmax is used as the vortex radius, thin lines and small markers correspondto usingr1 (see text).

expelled by these vortices (about 15%) is smaller, but comparable, to the mass expelled byRankine vortices. In contrast, the amount of mass transported in the exchange regime isconsiderably different: less than 20% in MMZ vortices and over 60% for Rankine vortices.

Instead of using the radius of zero vorticityr0 as a second definition of vortex radius, wewill use the radiusr1 where the vorticity reaches 1% of its peak value (note thatr1/r0 =0.83). If we define “filamentation” as the expulsion of filaments of fluid coming from theregionr < r1, then there are initial conditions in which vortices expel considerable amountsof filaments but do not merge (seeFig. 13). With the new definition the maximum values ofthe mass expelled during merger and of that transported during exchange rise considerably(to about 50 and 60%, respectively). Note also that the boundary between the exchange andthe elastic-interaction regimes change with the definition of the vortex radius.

5. Conclusions

Vortex filamentation has been reviewed using the theory of transport in dynamical sys-tems. From this viewpoint filamentation is a manifestation of lobe shedding; therefore, itoccurs because the stable manifold of a hyperbolic particle enters the vortex. The hyperbolictrajectory itself may, or may not, enter the vortex; and the same is true for any saddle point(bearing in mind that their locations depend on the reference frame).

The tendency towards axial symmetry of elliptic vortices increases with increasingsmoothness of the vorticity profile. For example, a Kirchhoff vortex with aspect ratioe = 3 preserves that aspect ratio during its evolution; vortices with a steep vorticity profile[n = 6 in (3)] change frome = 3 toe = 2, and vortices with smooth vorticity profile [n = 2in (3)] settle at arounde = 1.3. This decrease of aspect ratio is usually accompanied by


filamentation, but we have also observed that a small decrease of aspect ratio may occurwithout filamentation taking place.

The regimes of the vortex-pair interaction do not depend on the position of the vor-tices with respect to the separatrices of the Eulerian flow as proposed in previous works(seeMelander et al., 1988; Cerretelli and Williamson, 2003). Similarly, the regimes transi-tions are not related in an obvious way to changes in the topology of the Lagrangian flowgeometry.

In all simulations, merger was accompanied by filamentation, but in some of them fila-mentation did not lead to merger. It was also observed that, at least for vortices with step-likevorticity profile, merger starts before filamentation takes place. Filaments are thus not thecause of merger. In addition, it is important to note that even if merger were caused byfilamentation, this would not constitute an explanation for the occurrence of merger. Forthen it would remain to explain why circular vortices sometimes form filaments (and wouldmerge) and sometimes do not form filaments (and would not merge).

Acknowledgements

This research was done during a sabbatical stay at the Department of Mechanical andAerospace Engineering of UCSD. I gratefully acknowledge the hospitality of Professor PaulLinden and the financial support of the University of California Institute for Mexico andthe United States (UC MEXUS). I am also thankful for the comments of two anonymousreferees, which led to improvements of the paper.

Appendix A. Accuracy of the vortex-in-cell model

The accuracy of the simulations was monitored by computing the evolution of the fol-lowing (nominally preserved) quantities: (a) the integral moments of the vorticity distribu-tionMx = ∫

xω da,My = ∫yω da andMxy = ∫

(x2 + y2)ω da, (b) the kinetic energyK =1/2

∫(u2 + v2)da, and (c) the enstrophyE = ∫

ω2 da. As an example,Fig. A.1 shows thetime evolution of the error in the conservation of these properties for the unstable Kirchhoffvortex discussed in Section3. The error at timet is defined asX(t) = (X(t) −X(0))/X(0),whereX(t) is the value of the property at timet andX(0) is its value in the initial condi-tion. The smallest errors occur in the moments of the vorticity distribution, whereas thelargest ones occur in the enstrophy, which decreases by about 1.5% in the period discussedhere.

The numerical dissipation of the simulations at timet may be quantified by computingan effective Reynolds number (seeKoumoutsakos, 1997):

Reeff(t) = 4tΓ 2(t)

Mxy(t) −Mxy(0)

whereΓ is the total circulation of the flow. This gives aReeff of the order of 100,000 in thetimes discussed here.


Fig. A.1. Evolution of (nominally preserved) quantities during the evolution of a perturbed Kirchhoff vortexmodelled by the vortex-in-cell method (Fig. 3). (a) Moments of vorticity:Mx (dash-dotted line),My (dotted line),andMxy (continuous line); (b) total kinetic energyK; and (c) total enstrophyE. The horizontal axis gives the timein units of the rotation period of the unperturbed Kirchhoff vortex.

Therefore, it may be concluded that the vortex-in-cell model accurately computes theinviscid dynamics of 2D vortical structures, at least for the periods considered in the presentpaper.

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